function kap_return = KAPPA4(a, b, c, d, e, x0)  
 
    kap_return = zeros(size(x0));  
      
    y_prime = @(x) 4*a*x.^3 + 3*b*x.^2 + 2*c*x + d;  
    y_double_prime = @(x) 12*a*x.^2 + 6*b*x + 2*c;  
      
    for i = 1:length(x0)  
        x = x0(i);  
        y_prime_value = y_prime(x);  
        y_double_prime_value = y_double_prime(x);  
          
        if y_prime_value ~= 0 || y_double_prime_value ~= 0  
            kap_return(i) = abs(y_double_prime_value) / ((1 + y_prime_value^2)^(3/2));  
        else  
            kap_return(i) = 0;  
        end  
    end  
    